It is quite remarkable how high the failure rate for anti-abortion ballot initiatives is. Even in very conservative states (like South Dakota, for instance), voters seem to reliably reject any serious restrictions on abortion at the ballot box. Just about the only pro-life initiatives that have achieved any success are those dealing with parental involvement for minors.
From my observations of the last couple decades, I think it's clear that polling data isn't terribly accurate on this particular issue. While many Americans do continue to tell pollsters that they support restricting abortion, especially in later stages of pregnancy, not many of them actually vote for such restrictions when given the opportunity.